A Poincaré Inequality and Consistency Results for Signal Sampling on Large Graphs

Published: 16 Jan 2024, Last Modified: 16 Apr 2024ICLR 2024 spotlightEveryoneRevisionsBibTeX
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Keywords: large-scale graphs, signal sampling, graphons
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TL;DR: We formulate sampling problems in the graphon limit to discover intrinsic structures of large graph, with both theoretical guarantees and empirical evidence.
Abstract: Large-scale graph machine learning is challenging as the complexity of learning models scales with the graph size. Subsampling the graph is a viable alternative, but sampling on graphs is nontrivial as graphs are non-Euclidean. Existing graph sampling techniques require not only computing the spectra of large matrices but also repeating these computations when the graph changes, e.g., grows. In this paper, we introduce a signal sampling theory for a type of graph limit---the graphon. We prove a Poincaré inequality for graphon signals and show that complements of node subsets satisfying this inequality are unique sampling sets for Paley-Wiener spaces of graphon signals. Exploiting connections with spectral clustering and Gaussian elimination, we prove that such sampling sets are consistent in the sense that unique sampling sets on a convergent graph sequence converge to unique sampling sets on the graphon. We then propose a related graphon signal sampling algorithm for large graphs, and demonstrate its good empirical performance on graph machine learning tasks.
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Primary Area: general machine learning (i.e., none of the above)
Submission Number: 2914